An In-Place Sorting with O(n log n) Comparisons and O(n) Moves

We present the first in-place algorithm for sorting an array of size n that performs, in the worst case, at most O(n log n) element comparisons and O(n) element transports. This solves a long-standing open problem, stated explicitly, e.g., in [J.I. Munro and V. Raman, Sorting with minimum data movement, J. Algorithms, 13, 374-93, 1992], of whether there exists a sorting algorithm that matches the asymptotic lower bounds on all computational resources simultaneously

Radix Sorting With No Extra Space

It is well known that n integers in the range [1,n^c] can be sorted in O(n) time in the RAM model using radix sorting. More generally, integers in any range [1,U] can be sorted in O(n sqrt{loglog n}) time. However, these algorithms use O(n) words of extra memory. Is this necessary? We present a simple, stable, integer sorting algorithm for words of size O(log n), which works in O(n) time and uses only O(1) words of extra memory on a RAM model. This is the integer sorting case most useful in practice. We extend this result with same bounds to the case when the keys are read-only, which is of theoretical interest. Another interesting question is the case of arbitrary c. Here we present a black-box transformation from any RAM sorting algorithm to a sorting algorithm which uses only O(1) extra space and has the same running time. This settles the complexity of in-place sorting in terms of the complexity of sorting.Comment: Full version of paper accepted to ESA 2007. (17 pages

Idempotent permutations

Together with a characteristic function, idempotent permutations uniquely determine idempotent maps, as well as their linearly ordered arrangement simultaneously. Furthermore, in-place linear time transformations are possible between them. Hence, they may be important for succinct data structures, information storing, sorting and searching. In this study, their combinatorial interpretation is given and their application on sorting is examined. Given an array of n integer keys each in [1,n], if it is allowed to modify the keys in the range [-n,n], idempotent permutations make it possible to obtain linearly ordered arrangement of the keys in O(n) time using only 4log(n) bits, setting the theoretical lower bound of time and space complexity of sorting. If it is not allowed to modify the keys out of the range [1,n], then n+4log(n) bits are required where n of them is used to tag some of the keys.Comment: 32 page

Do Vouchers Lead to Sorting under Random Private School Selection? Evidence from the Milwaukee Voucher Program

This paper analyzes the impact of voucher design on student sorting, and more specifically investigates whether there are feasible ways of designing vouchers that can reduce or eliminate student sorting. It studies these questions in the context of the first five years of the Milwaukee voucher program. Much of the existing literature investigates the question of sorting where private schools can screen students. However, the publicly funded U.S. voucher programs require private schools to accept all students unless oversubscribed and to pick students randomly if oversubscribed. This paper focuses on two crucial features of the Milwaukee voucher program - random private school selection and the absence of topping up of vouchers. In the context of a theoretical model, it argues that random private school selection alone cannot prevent student sorting. However, random private school selection coupled with the absence of topping up can preclude sorting by income, although there is still sorting by ability. Sorting by ability is not caused here by private school selection, but rather by parental self selection. Using a logit model and student level data from the Milwaukee voucher program for 1990-94, it then establishes that random selection has indeed taken place so that it provides an appropriate setting to test the corresponding theoretical predictions in the data. Next, using several alternative logit specifications, it demonstrates that these predictions are validated empirically. These findings have important policy implications.Vouchers, Sorting, Cream Skimming, Private Schools